Variational and Potential Methods for a Class of Linear Hyperbolic Evolutionary Processes / Edition 1

by Igor Chudinovich

Overview

Variational and boundary integral equation techniques are two of the most useful methods for solving time-dependent problems described by systems of equations of the form 2— u = Au, 2—t 2 where u = u(x,t) is a vector-valued function, x is a point in a domain inR or 3 R,and A is a linear elliptic differential operator. To facilitate a better und- standing of these two types of methods, below we propose to illustrate their mechanisms in action on a specific mathematical model rather than in a more impersonal abstract setting. For this purpose, we have chosen the hyperbolic system of partial differential equations governing the nonstationary bending of elastic plates with transverse shear deformation. The reason for our choice is twofold. On the one hand, in a certain sense this is a “hybrid” system, c- sisting of three equations for three unknown functions in only two independent variables, which makes it more unusual—and thereby more interesting to the analyst—than other systems arising in solid mechanics. On the other hand, this particular plate model has received very little attention compared to the so-called classical one, based on Kirchho?’s simplifying hypotheses, although, as acknowledged by practitioners, it represents a substantial refinement of the latter and therefore needs a rigorous discussion of the existence, uniqueness, and continuous dependence of its solution on the data before any construction of numerical approximation algorithms can be contemplated.

Product Details

ISBN-13:	9781849969468
Publisher:	Springer London
Publication date:	12/13/2010
Series:	Springer Monographs in Mathematics
Edition description:	Softcover reprint of hardcover 1st ed. 2005
Pages:	148
Product dimensions:	6.10(w) x 9.25(h) x 0.01(d)

About the Author

Igor Chudinovich is Professor of Mathematics at the University of Guanajuato, Mexico, and Christian Constanda is Oliphant Professor of Mathematical Sciences at the University of Tulsa, USA.

Table of Contents

Formulation of the Problems and Their Nonstationary Boundary Integral Equations.- Problems with Dirichlet Boundary Conditions.- Problems with Neumann Boundary Conditions.- Boundary Integral Equations for Problems with Dirichlet and Neumann Boundary Conditions.- Transmission Problems and Multiply Connected Plates.- Plate Weakened by a Crack.- Initial-Boundary Value Problems with Other Types of Boundary Conditions.- Boundary Integral Equations for Plates on a Generalized Elastic Foundation.- Problems with Nonhomogeneous Equations and Nonhomogeneous Initial Conditions.

Editorial Reviews

"This book deals with variational and boundary integral equation techniques applied to hyperbolic systems of partial differential equations governing the nonstationary bending of elastic plates with transverse shear deformation. Results about the variational formulation are first established in the general setting of Sobolev spaces before useful, closed-form integral representations of the solutions are sought in terms of dynamic retarded plate potentials.

Most classical problems in the field of elastic plates are considered; namely, the clamped-edge and free-edge plates involving Dirichlet and Neumann boundary conditions respectively, elastic Robin, simply supported edge plates characterized by mixed and combined displacement-traction boundary data, transmission or contact problems, plates with homogeneous inclusions, cracks and plates on a generalized elastic foundation. The general scheme is the following: once the variational formulation has been established, the solvability is studied in distributional spaces; this allows the authors to determine solutions in the form of time-dependent single-layer and double-layer potentials with distributed densities satisfying nonstationary integral equations. The resulting equations are handled by means of the Laplace transform to reduce the original problems to boundary value problems depending on some complex transformation parameter $p$. When the real part of $p$ is constrained to be positive, estimates are derived, thus allowing the authors to make conclusions about the properties of solutions. To solve the transformed problems, algebras of singular integral operators are introduced so as to take into account the boundary values of the transformed potentials. Using Parseval's identity, the spaces of originals are recovered to prove existence of weak solutions to the given initial-boundary value problems...." (Isabelle Gruais, Mathematical Reviews)

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